Back to Dynamics of Machines

Study notes

LearnExamsInterviews

Qwestrum Engineering360 · Mechanical Engineering · Dynamics of Machines

Whirling of Shafts

A rotating shaft whirls violently when its speed equals its lateral natural frequency: the critical speed ω_cr = √(g/δ_st) = √(k/m). Operating well below (rigid) or above (flexible) ω_cr keeps deflection bounded, as SS Rattan explains.
Exam tip: use ω in rad/s for dynamics formulas; N in rpm needs ω = 2πN/60. Do not confuse flywheel C_s with governor speed control.

Key formulas & points

Skim these first — then read the full notes below.

  • Whirling: shaft bow rotates with unbalance at critical speed
  • Dunkerley's method estimates combined critical speeds
  • Gyroscopic effects raise/lower critical speeds of rotors

Topic details

Introduction

Whirling of shafts connects rotordynamics to vibration and is a frequent short question in Indian DOM papers. A shaft carrying a disc bows out and this bow rotates with the shaft; at the critical speed the centrifugal effect and elastic restoring force resonate, and deflection grows without bound.

Scope in B.Tech and GATE syllabus

SS Rattan shows the critical speed equals the transverse natural frequency of the shaft-disc system, ω_cr = √(k/m), which for a simply supported shaft with central load is √(g/δ_st). Dunkerley's method combines several discs' individual critical speeds into one estimate for the assembly.

Why this topic matters in practice

Practically, machines pass quickly through critical speed on run-up; steam turbines run above the first critical (flexible-shaft operation), while low-speed machines stay below it. Stating the safe margins (below 0.8ω_cr or above 1.2ω_cr) is expected.

Key relations & formulas

ωcr=(π2L2)EIm\omega_{cr} = (\frac{\pi^{2}}{L^{2}})\sqrt{\frac{EI}{m}}
(fundamental critical speed, simply supported; m = mass per unit length)
Ncr=30⋅ωcrπN_{cr} = 30\cdot \frac{\omega_{cr}}{\pi}
(rpm)
ωoperating<0.8⋅ωcr\omega_{operating} < 0.8\cdot \omega_{cr}
(safe below critical, rigid shaft)
ωoperating>1.2⋅ωcr\omega_{operating} > 1.2\cdot \omega_{cr}
(safe above critical, flexible shaft)

Notation and sign conventions

Relation 1 —
ωcr=\omega_{cr} =
ωcr=(π2L2)EIm\omega_{cr} = (\frac{\pi^{2}}{L^{2}})\sqrt{\frac{EI}{m}}
(fundamental critical speed, simply supported; m = mass per unit length)
Write this relation with symbols exactly as in SS Rattan — Theory of Machines before substituting numbers. Examiners award partial marks for a correct setup even when arithmetic slips.
Relation 2 —
Ncr=30⋅ωcrπN_{cr} = 30\cdot \frac{\omega_{cr}}{\pi}
Ncr=30⋅ωcrπN_{cr} = 30\cdot \frac{\omega_{cr}}{\pi}
(rpm)
Write this relation with symbols exactly as in SS Rattan — Theory of Machines before substituting numbers. Examiners award partial marks for a correct setup even when arithmetic slips.
Relation 3 —
ωoperating<0.8⋅ωcr\omega_{operating} < 0.8\cdot \omega_{cr}
ωoperating<0.8⋅ωcr\omega_{operating} < 0.8\cdot \omega_{cr}
(safe below critical, rigid shaft)
Write this relation with symbols exactly as in SS Rattan — Theory of Machines before substituting numbers. Examiners award partial marks for a correct setup even when arithmetic slips.
Relation 4 —
ωoperating>1.2⋅ωcr\omega_{operating} > 1.2\cdot \omega_{cr}
ωoperating>1.2⋅ωcr\omega_{operating} > 1.2\cdot \omega_{cr}
(safe above critical, flexible shaft)
Write this relation with symbols exactly as in SS Rattan — Theory of Machines before substituting numbers. Examiners award partial marks for a correct setup even when arithmetic slips.

Fundamentals and definitions

Model the shaft as a massless elastic beam carrying a disc of mass m at midspan; its lateral stiffness k gives natural frequency ω_n = √(k/m). Whirling occurs when the spin speed equals ω_n, so the critical speed ω_cr = ω_n.

Governing relations in practice

For a simply supported shaft with a central disc, static deflection δ_st = mgL³/48EI, and since k = mg/δ_st, ω_cr = √(g/δ_st). This convenient link lets the critical speed be found from the measured static sag.

Design and analysis considerations

At the critical speed any residual eccentricity e drives resonance and the dynamic deflection theoretically diverges (limited only by damping) — hence the destructive vibration. Above the critical speed the shaft self-centres: the disc rotates about its mass centre and deflection reduces.

Advanced theory and extensions

Dunkerley's empirical rule 1/ω_cr² = Σ1/ω_i² adds the reciprocal squares of individual critical speeds, giving a quick conservative estimate for multi-disc shafts. Gyroscopic effects of overhung discs shift the critical speeds and are noted in advanced treatments.

Assumptions and validity limits

State assumptions explicitly before using any relation for whirling of shafts — steady state, uniform properties, linear elastic material, ideal gas, incompressible flow, etc., as applicable.
Wrong assumptions invalidate the entire solution even when the formula is correct. In Dynamics of Machines viva and GATE descriptive questions, listing valid assumptions often earns separate marks.

Step-by-step problem approach

1. Read the question and list given data with SI units (common in Dynamics of Machines papers).
2. Draw a neat labelled diagram where applicable — examiners in Indian universities award diagram marks even when arithmetic slips.
3. Identify which relation from this topic applies to whirling of shafts.
4. Use equation 1:
ωcr=\omega_{cr} =
.
5. Use equation 2:
Ncr=30⋅ωcrπN_{cr} = 30\cdot \frac{\omega_{cr}}{\pi}
.
6. Substitute values, compute, and verify units and sign (direction).
7. State conclusion in one line — e.g. safe/unsafe, stable/unstable, feasible/infeasible.

Applications & exam relevance

Whirling of Shafts appears in engines, flywheels, and high-speed shafts. In Indian mechanical curricula this topic is tested because it connects theory to balancing, vibration, and rotational dynamics.
GATE and semester exams often combine whirling of shafts with earlier units — revise prerequisites before attempting mixed problems.
Industry interview panels sometimes ask: "Where did you use whirling of shafts?" — answer with a lab, mini-project, or plant visit example if possible.

Common mistakes in exams

• Confusing critical (whirling) speed with the torsional natural frequency of the shaft
• Using the wrong deflection formula δ_st for the given support/loading condition
• Assuming the shaft is unsafe above ω_cr — it self-centres and deflection actually reduces
• Forgetting Dunkerley combines reciprocals of squares, not the speeds directly

Quick revision checklist

Before attempting whirling of shafts problems, confirm you can:
1. Whirling: shaft bow rotates with unbalance at critical speed
2. Dunkerley's method estimates combined critical speeds
3. Gyroscopic effects raise/lower critical speeds of rotors
Revise the solved examples in SS Rattan — Theory of Machines and one previous-year GATE or university paper for this unit.

Worked examples

Try the problem first — open the solution when you are ready to check.

Critical speed from static deflection

Problem

A shaft with a central disc has a static deflection δ_st = 1 mm under the disc weight. Find the critical (whirling) speed in rad/s and rpm.

Solution

ω_cr = √(g/δ_st) = √(9.81/0.001) = √9810 = 99.0 rad/s; N_cr = 60ω_cr/2π = 60×99.0/6.283 = 946 rpm.

Conceptual check — Whirling of Shafts

Problem

In a Dynamics of Machines semester or GATE paper you are asked: "State the main assumption, the governing relation, and one practical consequence of whirling of shafts." What should a complete answer include?

Practice questions

Most-asked interview and GATE questions for this topic — expand any item for a model answer.

  1. 1
    What is Whirling of Shafts, and why does it appear in B.Tech / GATE syllabi?

    Model answer

    A rotating shaft whirls violently when its speed equals its lateral natural frequency: the critical speed ω_cr = √(g/δ_st) = √(k/m). Operating well below (rigid) or above (flexible) ω_cr keeps deflection bounded, as SS Rattan explains.
  2. 2
    State the relation ω_cr = and name each symbol.

    Model answer

    The governing relation is ωcr=\omega_{cr} =. Write every symbol with SI units before substituting numbers.
  3. 3
    State the relation N_cr = 30·ω_cr/π and name each symbol.

    Model answer

    The governing relation is Ncr=30⋅ωcrπN_{cr} = 30\cdot \frac{\omega_{cr}}{\pi}. Write every symbol with SI units before substituting numbers.
  4. 4
    State the relation ω_operating < 0.8·ω_cr and name each symbol.

    Model answer

    The governing relation is ωoperating<0.8⋅ωcr\omega_{operating} < 0.8\cdot \omega_{cr}. Write every symbol with SI units before substituting numbers.
  5. 5
    State the relation ω_operating > 1.2·ω_cr and name each symbol.

    Model answer

    The governing relation is ωoperating>1.2⋅ωcr\omega_{operating} > 1.2\cdot \omega_{cr}. Write every symbol with SI units before substituting numbers.
  6. 6
    Explain: Whirling: shaft bow rotates with unbalance at critical speed

    Model answer

    Whirling: shaft bow rotates with unbalance at critical speed — state the assumption range and one exam trap linked to this point.
  7. 7
    Explain: Dunkerley's method estimates combined critical speeds

    Model answer

    Dunkerley's method estimates combined critical speeds — state the assumption range and one exam trap linked to this point.
  8. 8
    Explain: Gyroscopic effects raise/lower critical speeds of rotors

    Model answer

    Gyroscopic effects raise/lower critical speeds of rotors — state the assumption range and one exam trap linked to this point.
  9. 9
    How would you correct this error in a viva: Confusing critical (whirling) speed with the torsional natural frequency of the shaft?

    Model answer

    Identify the wrong assumption or unit mix-up, rewrite the correct relation, and recompute with a one-line sanity check.
  10. 10
    How would you correct this error in a viva: Using the wrong deflection formula δ_st for the given support/loading condition?

    Model answer

    Identify the wrong assumption or unit mix-up, rewrite the correct relation, and recompute with a one-line sanity check.
  11. 11
    How would you correct this error in a viva: Assuming the shaft is unsafe above ω_cr — it self-centres and deflection actually reduces?

    Model answer

    Identify the wrong assumption or unit mix-up, rewrite the correct relation, and recompute with a one-line sanity check.
  12. 12
    How would you correct this error in a viva: Forgetting Dunkerley combines reciprocals of squares, not the speeds directly?

    Model answer

    Identify the wrong assumption or unit mix-up, rewrite the correct relation, and recompute with a one-line sanity check.

Exams & GATE

  • 1
    Never operate at critical speed — resonance causes catastrophic failure.
  • 2
    Avoid: Confusing critical (whirling) speed with the torsional natural frequency of the shaft
  • 3
    Avoid: Using the wrong deflection formula δ_st for the given support/loading condition
  • 4
    Avoid: Assuming the shaft is unsafe above ω_cr — it self-centres and deflection actually reduces

📖 Standard books (India)

  • Theory of Machines — SS Rattan

    Read: Syllabus unit

    Kinematics, cams, governors, and balancing

Explore related topics

See real mechanical engineering careers

After exams and interviews, see how engineers actually built careers — milestones and decisions from people in the field.

Browse Mechanical Engineering journeysPlan your path